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href="chap1.html" >1</a> <a href="chap2.html" >2</a> <a href="chap3.html" >3</a> <a href="chap4.html" >4</a> <a href="chapA.html" >A</a> <a href="chapB.html" >B</a> <a href="chapBib.html" >Bib</a> <a href="chapInd.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap4.html" >[Previous Chapter]</a> <a href="chapB.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chapA_mj.html" >[MathJax on]</a></p>"X7B2993CB7B012115" name="X7B2993CB7B012115" ></a></p>div class="ChapSects" ><a href="chapA.html#X7B2993CB7B012115" >A <span class="Heading" >The Matrix Tool Operations</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapA.html#X7988F0AF7D87FD23" >A.1 <span class="Heading" >The Tool Operations <em >without</em > a Fallback Method</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chapA.html#X7912E42C81296637" >A.2 <span class="Heading" >The Tool Operations with a Fallback Method</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapA.html#X84C78C6B7C0890BF" >A.2-1 MonomialMatrix</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapA.html#X7F9F9C978703E871" >A.2-2 RandomMatrixBetweenGradedFreeLeftModules</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapA.html#X82E813D982331C0B" >A.2-3 RandomMatrixBetweenGradedFreeRightModules</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chapA.html#X7EA1DC07822DE322" >A.2-4 Diff</a></span >div ></div >div >span class="Heading" >The Matrix Tool Operations</span ></h3>code class="code" >homalgTable</code > object stored in the ring. They are only indirectly accessible through standard methods that invoke them.</p>"X7988F0AF7D87FD23" name="X7988F0AF7D87FD23" ></a></p>span class="Heading" >The Tool Operations <em >without</em > a Fallback Method</span ></h4>strong class="pkg" >homalg</strong > needs a <code class="code" >homalgTable</code > entry for non-internal rings, as it cannot provide a suitable fallback. Below is the list of these <code class="code" >homalgTable</code > entries.</p>"X7912E42C81296637" name="X7912E42C81296637" ></a></p>span class="Heading" >The Tool Operations with a Fallback Method</span ></h4>code class="code" code > entry for non-internal rings. <strong class="pkg" >homalg</strong > only provides a generic fallback method.</p>"X84C78C6B7C0890BF" name="X84C78C6B7C0890BF" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ MonomialMatrix</code >( <var class="Arg" >d</var >, <var class="Arg" >R</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > matrix</p>var class="Arg" >d</var >-th monomials of the <strong class="pkg" >homalg</strong > graded ring <var class="Arg" >R</var >.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z" ;;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >S := GradedRing( R );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >m := MonomialMatrix( 2, S );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >NumberRows( m );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >m;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Display( m );</span >pre ></div >"X7F9F9C978703E871" name="X7F9F9C978703E871" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ RandomMatrixBetweenGradedFreeLeftModules</code >( <var class="Arg" >degreesS</var >, <var class="Arg" >degreesT</var >, <var class="Arg" >R</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div strong class="pkg" >homalg</strong > matrix</p>span class="SimpleMath" >r × c</span >-matrix between the graded free <em >left</em > modules <span "SimpleMath" ><var class="Arg" >R</var >^(-<var class="Arg" >degreesS</var >) -> <var class="Arg" var >^(-<var class="Arg" >degreesT</var >)</span >, where <span class="SimpleMath" >r =</span ><code class="code" >Length</code ><span class="SimpleMath" >(</span ><var class="Arg" >degreesS</var ><span class="SimpleMath" >)</span > and <span class="SimpleMath" >c =</span ><code class="code" >Length</code ><span class="SimpleMath" >(</span ><var class="Arg" >degreesT</var ><span class="SimpleMath" >)</span >.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c" ;;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >S := GradedRing( R );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >rand := RandomMatrixBetweenGradedFreeLeftModules( [ 2, 3, 4 ], [ 1, 2 ], S );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >#Display( rand );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >#a-2*b+2*c, 2, </span >span class="GAPprompt" >gap></span > <span class="GAPinput" >#a^2-a*b+b^2-2*b*c+5*c^2, 3*c, </span >span class="GAPprompt" >gap></span > <span class="GAPinput" >#2*a^3-3*a^2*b+2*a*b^2+3*a^2*c+a*b*c-2*b^2*c-3*b*c^2-2*c^3,a^2-4*a*b-3*a*c-c^2</span >pre ></div >"X82E813D982331C0B" name="X82E813D982331C0B" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ RandomMatrixBetweenGradedFreeRightModules</code >( <var class="Arg" >degreesT</var >, <var class="Arg" var >, <var class="Arg" >R</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > matrix</p>span class="SimpleMath" >r × c</span >-matrix between the graded free <em >right</em > modules <span class="SimpleMath" ><var class="Arg" >R</var >^(-<var class="Arg" >degreesS</var >) -> <var class="Arg" >R</var >^(-<var class="Arg" >degreesT</var >)</span >, where <span class="SimpleMath" >r =</span ><code "code" >Length</code ><span class="SimpleMath" >(</span ><var class="Arg" >degreesT</var ><span "SimpleMath" >)</span > and <span class="SimpleMath" >c =</span ><code class="code" >Length</code ><span class="SimpleMath" >(</span ><var class="Arg" >degreesS</var ><span class="SimpleMath" >)</span >.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c" ;;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >S := GradedRing( R );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >rand := RandomMatrixBetweenGradedFreeRightModules( [ 1, 2 ], [ 2, 3, 4 ], S );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >#Display( rand );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >#a-2*b-c,a*b+b^2-b*c,2*a^3-a*b^2-4*b^3+4*a^2*c-3*a*b*c-b^2*c+a*c^2+5*b*c^2-2*c^3,</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >#-5, -2*a+c, -2*a^2-a*b-2*b^2-3*a*c </span >pre ></div >"X7EA1DC07822DE322" name="X7EA1DC07822DE322" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ Diff</code >( <var class="Arg" >D</var >, <var class="Arg" >N</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > matrix</p>var class="Arg" >D</var > is a <span class="SimpleMath" >f × p</span >-matrix and <var class="Arg" >N</var span class="SimpleMath" >g × q</span >-matrix then <span class="SimpleMath" >H=Diff(</span ><var class="Arg" >D</var >,<var class="Arg" >N</var ><span class="SimpleMath" >)</span > is an <span class="SimpleMath" >fg × pq</span >-matrix whose entry <span class="SimpleMath" >H[g*(i-1)+j,q*(k-1)+l]</span > is the result of differentiating <var class="Arg" >N</var ><span class="SimpleMath" >[j,l]</span > by the differential operator corresponding to <var class="Arg" >D</var ><span class="SimpleMath" >[i,k]</span >. (Here we follow the Macaulay2 convention.)</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >S := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c" * "x,y,z" ;;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >D := HomalgMatrix( "[ \ span class="GAPprompt" >></span > <span class="GAPinput" >x,2*y, \</span >span class="GAPprompt" >></span > <span class="GAPinput" >y,a-b^2, \</span >span class="GAPprompt" >></span > <span class="GAPinput" >z,y-b \</span >span class="GAPprompt" >></span > <span class="GAPinput" >]", 3, 2, S ); span class="GAPprompt" >gap></span > <span class="GAPinput" >N := HomalgMatrix( "[ \ span class="GAPprompt" >></span > <span class="GAPinput" >x^2-a*y^3,x^3-z^2*y,x*y-b,x*z-c, \</span span class="GAPprompt" >></span > <span class="GAPinput" >x, x*y, a-b, x*a*b \</span >span class="GAPprompt" >></span > <span class="GAPinput" >]", 2, 4, S ); span class="GAPprompt" >gap></span > <span class="GAPinput" >H := Diff( D, N );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Display( H );</span >pre ></div >div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap4.html" >[Previous Chapter]</a> <a href="chapB.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1</a> <a href="chap2.html" >2</a> <a href="chap3.html" >3</a> <a href="chap4.html" >4</a> <a href="chapA.html" >A</a> <a href="chapB.html" >B</a> <a href="chapBib.html" >Bib</a> <a href="chapInd.html" >Ind</a> </div >hr />"foot" >generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html > Messung V0.5 C=89 H=100 G=94
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